A Very Efficient Time and Frequency Synchronization Method for OFDM Systems Operating in AWGN Channels Leïla Nasraoui, Leïla Najjar Atallah
Mohamed Siala
TECHTRA Laboratory Higher School of Communications University of 7 November at Carthage, Tunisia.
[email protected],
[email protected]
MEDIATRON Laboratory Higher School of Communications University of 7 November at Carthage, Tunisia.
[email protected]
Abstract— In this paper, we propose an efficient scheme using a new single-symbol preamble structure for OFDM data-aided synchronization. The preamble symbol is constituted of two successive equal-duration sub-symbols, the second one of which is a differentially encoded version of the first one, using an adequately designed precoding sequence. This structure leads to almost perfect channel autocorrelation profile with sharp inphase peak and zero out-of-phase correlation. For comparison, we review some previous methods also based on data-aided synchronization for OFDM systems. We here explore the performance of the new method in the case of the AWGN channel. The performance is presented in terms of correct frame start detection rate and estimator’s variance for time estimation. For fractional frequency offset estimation, the performance is evaluated in terms of mean squared error. The obtained results prove that the accuracy of the frame start detection and the fractional frequency offset estimation are greatly enhanced, even at very low SNRs. We show that compared to the best benchmark, gains of about 5 dB and 20 dB are achieved for timing precision of one sample and of one tenth of a sample, respectively. We also show that for a target correct detection rate of 95%, our approach provides more than 17 dB gain with respect to the best benchmark. Although we concentrate here on OFDM, the proposed approach can be applied to other systems such as TDMA or CDMA. Keywords- OFDM; preamble-based synchronization; precoding; preferred pair of m-sequence; time synchronization; frequency synchronization; preamble sequence design
I.
INTRODUCTION
Orthogonal Frequency Division Multiplexing (OFDM) systems have improved both wired and wireless communications thanks to their high data rate transmission and robustness to channel’s imperfections. A misalignment between the sent symbol start and the demodulated one (timing error) can introduce Inter-Symbol Interference (ISI). The use of a Cyclic Prefix (CP) longer than the channel delay spread renders the OFDM systems robust to timing offsets lower than the CP excess length. As the CP length is generally chosen to sensibly equal the channel delay spread, the timing offset must be accurately recovered to avoid the ISI. A frequency error results in Inter-Carrier Interference (ICI) that destroys the
orthogonality between sub-carriers. The lower is the subcarriers spacing, the more is the sensitivity to frequency offsets. The timing and frequency offsets deteriorate modulation performance. To face those limitations, many approaches have been proposed in the literature to estimate the time and/or the frequency offsets. Some of them exploit the redundancy in the CP, as the work of Van de Beek in [3], for blind synchronization, which is more suited to continuous flow transmission, such as streaming video. Others use a preamble before OFDM symbols data transmission, as in [1] and [2]. Those approaches, known as data-aided synchronization methods, cost in terms of bandwidth but are generally more efficient than blind ones, especially for bursty packet traffic. In between, we have semi-blind synchronization that uses the redundancy in the CP in addition to pilot symbols sent regularly for channel estimation, as presented in [4]. This paper addresses the problem of data-aided synchronization. It is organized as follows. Section II briefly defines the OFDM signal. In section III, we study some existing synchronization methods presented in [1] and [2], with the intention of using them as benchmarks. Then, we describe our synchronization approach, in section IV. Simulations results and discussions are given, in section V. Finally, our conclusions and perspectives are provided in Section VI. II.
OFDM SIGNAL
The OFDM signal is generated by modulating equally spaced sub-carriers, using QAM or PSK complex data symbols. These symbols denoted ck , n are indexed by the OFDM symbol number k and the sub-carrier number n (n = 0,1,..., N u − 1). the transmitted baseband OFDM signal after pulse shape filtering is given by Nu
s ( t ) = ∑ ck , n e n =0
j 2 π f n ( t − kT −Tg )
, t ∈ [ kT , ( k + 1)T ],
(1)
th
where N u is the number of sub-carriers. The n sub-carrier frequency is denoted by f n = n / Tu , where Tu is the length of
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the OFDM symbol useful part. The considered guard interval extension consists in appending a copy of the last Tg period of
where a is a Pseudo Noise (PN) sequence of length L = N u / 2 . To estimate the frame start, Schmidl and Cox take
the OFDM symbol as a prefix. T = Tu + Tg denotes the whole
the maximum point d opt of the timing metric given by
OFDM symbol duration. The OFDM signal discrete formulation with sampling period Ts = Tu / N u coincides, for
M 1 (d ) =
the k OFDM symbol, with the N u points Inverse Fast Fourier th
{c } .
Transform (IFFT) of
k ,n
The added CP of length
N g = Tg / Ts is here directly generated in the time domain (no
P1 ( d )
2
(4)
( R1 ( d )) 2
where is the time index corresponding to the first sample in a sliding window of N u samples,
th
IFFT is used). The k transmitted samples of the OFDM symbol including CP are given by
skN + l =
j 2π ( l − N g )
N u −1
∑c
k ,n
e
Nu
n=0
, l ∈ [ 0, N − 1] , k ∈ Z,
(2)
L −1
P1 (d ) = ∑ rd*+ m rd + m + L , and L −1
R1 ( d ) = ∑ rd + m + L .
where N = N u + N g .
Due to miss-synchronized transmitter and receiver oscillators, the received signal is affected by a frequency offset, the normalized version of which, with respect to sub-carriers spacing, is denoted by υ . Moreover, the received signal has a time offset, the normalized version of which, with respect to the sampling period is denoted by τ . The channel introduces a zero-mean Additive White Gaussian Noise (AWGN) ωk
(k ∈ Z). The received sample expression is then
rk = e
j 2 πυ
k Nu
sk −τ + ωk .
(3)
In the following, we consider a class of preamble-based synchronization methods that use a preamble, with specific structure for synchronization aims. This preamble is sent either periodically, to track the synchronization parameters, in the case of continuous stream traffic or at the start of a transmission, to detect the beginning of emission, in the case of bursty packet traffic. III.
SURVEY ON PREAMBLE-BASED SYNCHRONIZATION METHODS
This section introduces three methods applied for preamble-based OFDM synchronization. The first method, proposed by Schmidl and Cox, suggests two estimators, for Time Offset (TO) and Fractional Frequency Offset (FFO) [1]. The second, known as Minn’s Sliding Window Method, is a modified version of the previous TO estimator [2]. Finally, the third method, named Minn’s Training Symbol Method, presents a different TO estimator using a different preamble structure [2]. A. Schmidl and Cox method Schmidl and Cox have presented in [1] a robust frequency and timing synchronization algorithm. The preamble is composed of two symbols. Only the first is used for TO and FFO estimation. This symbol is designed to be
Ps &c = [ a , a ] ,
(5)
m =0
2
(6)
m =1
The main drawback of this approach is that the metric exhibits a plateau which has a length equal to that of the guard interval minus the length of the channel impulse response. Due to noise, this plateau leads to some uncertainty as to the start of the frame and results in a large TO estimation variance. To alleviate this drawback, Schmidl and Cox proposed an averaging method, where two points to the left and right in the time domain, which are exactly within 90% of the maximum value, are first found. Then, the timing estimate is taken as the middle of those two points. In this way, the estimate is guaranteed to fall within the plateau. The metric in (5) is also used for frequency synchronization to detect the FFO by
1 υ = (( P1 ( d opt )) Tuπ
(7)
where ( denotes the phase operator. To further reduce the uncertainty due to the timing metric plateau, two approaches have been proposed by Minn et al. in [2]. These methods are presented hereafter. B. Minn’s Sliding Window Method This method uses the same preamble as the method presented in subsection A. Two aspects have been changed: •
The first modification consists in computing the energy R1 ( d ) over the whole symbol, rather than over the second half. This leads to
R2 (d ) =
1N
−1
∑r 2 u
d +m
2
.
(8)
m =0
•
The second modification is averaging the metric over a window of length N g + 1 instead of the 90% averaging approach presented above.
The modified timing metric is given by
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M 2 (d ) =
1 Ng
0
∑M +1
( d + k ), f
(9)
k =− Ng
where Mf =
P1 ( d )
th
2
( R2 ( d ))
2
.
(10)
This approach uses a different preamble structure, designed
PMinn = [ b, b, −b, −b ] , where b is a sub-sequence of L = N u / 4 samples generated by taking the L = N u / 4 points IFFT of a PN sequence of length L. The time metric M 3 has the same form as in (4), with functions P1 and R1 modified respectively by P3 and R3 given by 1
L −1
P3 ( d ) = ∑∑ r
* d + 2 kL + m d + 2 kL + m + L
r
,
obtained by differentially modulating the first sub-sequence s1 using a precoding sequence p as follows
N um 2
− 1,
(13)
where N um is the length of the preamble (except the CP), s2,k and
s1,k are respectively the k th samples of sequences s1 and s2 1
L −1
R3 (d ) = ∑∑ rd + 2 Lk + m + L . 2
(12)
Illustrative curves representing the metrics of Schmidl and Cox’s method and Minn’s methods are shown in Fig. 1. The Sliding Window Method of Minn is referenced as “Method A”, and the Training Symbol Method of Minn is referenced as “Method B”. We here consider the case of an OFDM signal with N u = 1024 sub-carriers, with a guard interval of
N g = 102 samples, and a normalized carrier frequency offset υ = 12.4 sub-carriers spacing, and an ideal channel case (no
noise).
Proposed method Schmidl & Cox Minn Method A Minn Method B
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 100
200
300
400 500 600 Time (in samples)
700
th
and pk is the k sample of the precoding sequence. Each of these sequences ( s1 , s2 , and p) have a length of Lm = 2 − 1 q
k =0 m =0
Metric Mi, i=1,2,3,4
A. Proposed preamble structure The proposed preamble is composed of a single timedomain OFDM symbol. Abstraction made of the CP, this preamble is constructed as the concatenation of two equallength sub-sequences s1 and s2 . The second sub-sequence s2 is
(11)
and
0
PROPOSED SYNCHRONIZATION METHOD
Like the method presented in [5], we here use a differential coding in the time domain for the preamble design. We focus on the choice of the sub-sequences involved in the preamble construction.
s2, k = s1, k pk , k = 0,1,..., Lm − 1 =
k =0 m=0
0
since the preamble is sent at the 460 sample before which random data were sent. IV.
C. Minn’s Training Symbol Method as
It is shown that the plateau effect disappears in Minn’s methods. Nevertheless, the obtained correlation peak is not sharp leading to reduced robustness to noise. There is one maximum point defining the frame start. In this case, it th coincides with the 563 sample as can be seen in the figure,
800
Figure 1. Timing metrics under noiseless conditions.
900
1000
q ∈ ` ). The concatenated preamble samples form the sequence s = [ s1 , s2 ] = [ s1 , s1 : p ] where : denotes the
(where
element-wise product operator. To observe the OFDM symbol structure, a CP of length N gm = N g + N u − N um is added to
to keep the total OFDM symbol length of N . The choice of the sequences p and s1 is of utmost importance basic step. These sequences should be meticulously chosen to have an important correlation property, that will be exploited later when calculating the metric. M-sequences are well known for having specific correlation properties in terms of minimizing the maximum value of the out-of-phase autocorrelation. The smaller is the out-of-phase autocorrelation of the chosen sequence, the more accurate is the timing offset estimation. Thus, they are adequate for synchronization. Hence, in this work two m-sequences are applied to generate the time domain preamble ( s1 and p ). For a judicious choice, we use a preferred pair of m-sequences for which the above mentioned correlation property is better verified than arbitrary chosen subsequences. B. Timing offset estimation To detect the frame start, the idea is to search for two strongly correlated symbol halves. This is realized by searching the maximum of metric
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M4 =
P4 ( d )
2
( R4 ( d )) 2
,
(14)
where
P4 =
performance is evaluated over an SNR range from −10 dB to 25 dB which encompasses all practical operating values. B. Timing estimation performance
Lm −1
∑r m =1
*
*
r
d + m d + m + Lm
p m,
(15)
and
R4 (d ) =
Lm −1
∑r m =0
2
d + m + Lm
.
(16)
The correlation property of the chosen sequences ensures that the proposed timing metric has its peak value at the correct symbol starting point. The frame start corresponds to the sample index d opt that maximizes the metric in (14). Fig .1 shows that this metric exhibits an almost perfect channel autocorrelation profile with a sharp peak at the correct frame start, using a system of N um = 1022 samples and a guard interval of N gm = 104 samples.
The timing estimator performance is evaluated in terms of rate of correct detection and estimator variance. The five cases presented in [2] (I-Schmidl and Cox, II-Schmidl and Cox using (8), III-Minn’s Sliding Window Method, IV-Minn’s Training Symbol Method, and V-Minn’s Training Symbol Method using (8)) are compared to the proposed method. Fig. 2 shows the rate of correct detection for all methods. As can be noticed, beyond an SNR of -5 dB, the proposed method provides a correct start frame detection with rate 1. At very low SNR (less than 0 dB) method III outperforms the others. At higher SNR, the detection rates of methods IV and V are better than III, as the metric peak can be more easily distinguished. In all cases, methods I and II give the worst detection rates due to the plateau effect. A gain of about 15 dB is achieved by the proposed method with respect to the best benchmark (method V) for a target correct detection rate of 90%. This gain goes to 17 dB for a target of 95%.
C. Fractional frequency offset estimation
υˆ =
1
π
( ( P4 ( d opt )).
(17)
This equation provides a detection range of ±1 in terms of subcarriers spacing due to the phase of P4 , being unambiguously determined for −π < ( ( P4 ( d opt )) ≤ π . V.
1 0.9 0.8 Rate of Correct Detection
The correlation function in (15) used for timing synchronization is also exploited to calculate the FFO. Reaching the pick, the phase of P4 is πυ . Hence, to enhance υ estimation robustness to noise, we evaluate the phase of at the optimum timing d opt such that
0.7
Schmidl & Cox (I) Schmidl & Cox modified (II) Minn Method A (III) Minn Method B (IV) Minn Method B modified (V) Minn B modified (V) Proposed method
0.6 0.5 0.4 0.3 0.2
SIMULATIONS AND PERFORMANCE EVALUATION
The performance of the proposed synchronization method and the methods presented in section III are investigated by Matlab simulations.
0.1 0 -10
-5
0
5 10 SNR (dB)
15
20
25
Figure 2. Rate of correct detection of symbol timing.
A. Simulations parameters To be fair on comparing our method to the other benchmark methods exposed above, we have approximately chosen the same system parameters. Thus, we have used a preferred pair of 9 m-sequences of length N um = 2 − 1 = 511, and the guard interval N gm is set to 104 samples. For the other methods N u is chosen equal to 1024 and the guard interval length N g is equal to 102 samples. The guard interval of the proposed method has two samples more than the others to keep the same OFDM symbol length N = 1126 samples. The carrier frequency offset is set to 12.4 sub-carriers spacing. We treat the case of an 4
AWGN channel. For all simulations, between 10 and
5.106 trials are run, depending on the case at hand. The
Fig. 3 shows the variance of the TO estimator. At a very low SNR (around -10 dB) the performance of all the considered methods are comparable. For SNR higher than -5 dB, the proposed method leads to a variance equal to 0 (for 5.106 trials). This is coherent with the correct start frame detection rate shown before (Fig. 2). Other estimators variances’ go on declining. Methods I and II show a floor, while methods IV and V keep on decreasing. Note that methods using (8) have smaller variance. For a precision of one sample (var(τ ) = 1), a gain of about 5 dB is obtained compared to the best benchmark (method V). This gain increases to 20 dB for a precision of one tenth of a sample ( (var(τ ) = 10 ) ). Under -6 dB, the proposed method leads to slightly worse performance. We can think of applying an hybrid method where the best performing technique, namely method B of Minn in this SNR range.
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−2
C. Fractional frequency offset estimation performance For fractional frequency estimation, we set the normalized FFO υ to 0.4 sub-carriers spacing. To evaluate its estimation performance, we use the criterion of Mean Squared Error (MSE). The obtained simulation results are presented in Fig .4. As a second step in our evaluation process, we examine, for each method, the degradation effect of a time estimation error on FFO estimation performance. We simulate (5) and (15) for two time values: The perfect (correct) timing value and the timing value that maximizes the metrics, respectively given by (4) and (14), for Schmidl and Cox and the proposed scheme. In Fig .4, they are annotated respectively by I (correct) and II (estimated). Using the correct symbol timing, the proposed method gives better results when the SNR is less than -5 dB. Beyond this value, performance for true and estimated frame starting point are similar. The FFO estimation is robust to small timing offsets. For our method, this is obtained thanks to its sharp metric around the true frame start allowing a correct timing acquisition. For the method of Schmidl and Cox, the FFO estimation is insensitive to timing errors whenever the estimated timing lies within the plateau. We expect a degradation in multipath operation mode because the plateau is shortened by the dispersive nature of the channel. Compared to Schmidl and Cox’s method, the proposed one gives lower MSE.
Simulations results showed that the proposed method gives very accurate estimates of time and fractional frequency offsets. We have proved that, compared to the best benchmark, a gain of about 20 dB is achieved for timing precision of one tenth of a sample. For a target correct detection rate of 95%, our approach provides more than 17 dB gain, with respect to the best benchmark. Ongoing work aims to explore the proposed scheme performance in the case of multipath channels and to extend it to include integer frequency offset estimation. 0
10
Schmidl & Cox (I) Schmidl & Cox (II) Proposed Method (I) Proposed Method (II)
-1
10
-2
10
MSE(ν)
However, since there is no universal structure for the preamble, which is compliant with both methods, it is not possible to use any hybrid technique.
-3
10
-4
10
-5
10
-6
10
-10
-5
0
[1] 4
10
[2] 3
10
Schmidl & Cox (I)
[3]
Schmidl & Cox modified (II)
2
10
Minn Method A (III) var(τ)
15
20
25
REFERENCES
5
Minn Method B (IV)
1
10
Minn Method B modified (V)
[4]
Proposed method 0
10
[5]
-1
10
-2
10
T.M. Schmidl, D. Cox, “Robust frequency and timing synchronisation in OFDM,” IEEE Trans. on Comm., vol. 45, Dec. 1997, pp. 1613-1621. H. Minn, M. Zeng, V. K. Bhargava, “On timing offset estimation for OFDM systems,” IEEE Comm. Letters, vol.4, no. 7, July 2000, pp. 242244. J.-J. van de Beek, M. Sandell, M. Isaksson, and P. Börjesson, “Low complex frame synchronization in OFDM systems,” in Proc. ICUPC, Nov. 6–10, 1995, pp. 982–986. D. Landström, S. K. Wilson, J. J. van de Beek, P. Ödling, and P. O. Börjesson, “Symbol time offset estimation in coherent OFDM systems,” in Proc. Int. Conf. on Communications, Vancouver, BC, Canada, June1999, pp. 500–505. L. Atallah, M. Siala, “A New scheme for preamble detection and frequency acquisition in OFDM systems,” in Proc. ICECS, Hammamet, Tunisia, December 2009.
-3
-10
10 SNR (dB)
Figure 4. MSE of FFO for perfect and estimated symbol timing.
10
10
5
-5
0
5
10
15
20
25
SNR (dB)
Figure 3. Timing offset estimator variance.
VI.
CONCLUSION
A data-aided time and frequency synchronization method for OFDM systems has been presented. It is based on a new preamble structure that uses one symbol whose two halves are related by a differential encoding. A judicious design of the involved sequences guarantees a robust metric shape which enhances the frame start detection capacity and the FFO estimation accuracy.
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